A generalization of conjugation of integer partitions
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arXiv
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| Autori principali: | , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Albion, Seamus Eisenkölbl, Theresia Fischer, Ilse Gangl, Moritz Höngesberg, Hans Krattenthaler, Christian Rubey, Martin |
| author_facet | Albion, Seamus Eisenkölbl, Theresia Fischer, Ilse Gangl, Moritz Höngesberg, Hans Krattenthaler, Christian Rubey, Martin |
| contents | We exhibit, for any positive integer parameter $s$, an involution on the set of integer partitions of $n$. These involutions show the joint symmetry of the distributions of the following two statistics. The first counts the number of parts of a partition divisible by $s$, whereas the second counts the number of cells in the Ferrers diagram of a partition whose leg length is zero and whose arm length has remainder $s-1$ when dividing by $s$. In particular, for $s=1$ this involution is just conjugation. Additionally, we provide explicit expressions for the bivariate generating functions.
Our primary motivation to construct these involutions is that we know only of two other "natural" bijections on integer partitions of a given size, one of which is the Glaisher-Franklin bijection sending the set of parts divisible by $s$, each divided by $s$, to the set of parts occurring at least $s$ times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16043 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalization of conjugation of integer partitions Albion, Seamus Eisenkölbl, Theresia Fischer, Ilse Gangl, Moritz Höngesberg, Hans Krattenthaler, Christian Rubey, Martin Combinatorics Primary 05A19, Secondary 05A15 05A17 05A30 We exhibit, for any positive integer parameter $s$, an involution on the set of integer partitions of $n$. These involutions show the joint symmetry of the distributions of the following two statistics. The first counts the number of parts of a partition divisible by $s$, whereas the second counts the number of cells in the Ferrers diagram of a partition whose leg length is zero and whose arm length has remainder $s-1$ when dividing by $s$. In particular, for $s=1$ this involution is just conjugation. Additionally, we provide explicit expressions for the bivariate generating functions. Our primary motivation to construct these involutions is that we know only of two other "natural" bijections on integer partitions of a given size, one of which is the Glaisher-Franklin bijection sending the set of parts divisible by $s$, each divided by $s$, to the set of parts occurring at least $s$ times. |
| title | A generalization of conjugation of integer partitions |
| topic | Combinatorics Primary 05A19, Secondary 05A15 05A17 05A30 |
| url | https://arxiv.org/abs/2407.16043 |